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MATH 237 Differential Equations and Computer Methods

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MATH 237 Differential Equations and Computer Methods Queen’s University Mathematics and Engineering and Mathematics and Statistics MATH 237 Differential Equations and Computer Methods Supplemental Course Notes Serdar Y¨uksel November 19, 2010 This document is a collection of supplemental lecture notes used for Math 237: Differential Equations and Computer Methods. Serdar Y¨uksel Contents 1 Introduction to Differential Equations 7 1.1 Introduction: ................................... ....... 7 1.2 Classification of Differential Equations . ............... 7 1.2.1 OrdinaryDifferentialEquations. .......... 8 1.2.2 PartialDifferentialEquations . .......... 8 1.2.3 Homogeneous Differential Equations . .......... 8 1.2.4 N-thorderDifferentialEquations . ......... 8 1.2.5 LinearDifferentialEquations . ......... 8 1.3 Solutions of Differential equations . .............. 9 1.4 DirectionFields................................. ........ 10 1.5 Fundamental Questions on First-Order Differential Equations............... 10 2 First-Order Ordinary Differential Equations 11 2.1 ExactDifferentialEquations. ........... 11 2.2 MethodofIntegratingFactors. ........... 12 2.3 SeparableDifferentialEquations . ............ 13 2.4 Differential Equations with Homogenous Coefficients . ................ 13 2.5 First-Order Linear Differential Equations . .............. 14 2.6 Applications.................................... ....... 14 3 Higher-Order Ordinary Linear Differential Equations 15 3.1 Higher-OrderDifferentialEquations . ............ 15 3.1.1 LinearIndependence . ...... 16 3.1.2 WronskianofasetofSolutions . ........ 16 3.1.3 Non-HomogeneousProblem . ..... 17 4 Higher-Order Ordinary Linear Differential Equations 19 4.1 Homogeneous Linear Equations with Constant Coefficients ................. 19 3 4.1.1 L(m)withdistinctRealroots. 19 4.1.2 L(m)withrepeatedRealroots ............................ 20 4.1.3 L(m)withComplexroots ............................... 20 4.2 Non-Homogeneous Equations and the Principle of Superposition ............. 20 4.2.1 MethodofUndeterminedCoefficients . ........ 21 4.2.2 VariationofParameters . ....... 22 4.2.3 ReductionofOrder.............................. ..... 23 4.3 Applications of Second-Order Equations . ............. 24 4.3.1 Resonance..................................... 25 5 Systems of First-Order Linear Differential Equations 27 5.0.2 Higher-Order Linear Differential Equations can be reduced to First-Order Systems 28 5.1 TheoryofFirst-OrderSystems . ......... 29 6 Systems of First-Order Constant Coefficient Differential Equations 31 6.1 HomogeneousCase................................. ...... 31 6.2 How to Compute the Matrix Exponential . .......... 32 6.3 Non-HomogeneousCase .............................. ...... 33 6.4 RemarksontheTime-VaryingCase . ........ 34 7 Stability and Lyapunov’s Method 35 7.1 Stability....................................... ...... 35 7.1.1 LinearSystems ................................. 35 7.1.2 Non-LinearSystems .............................. 36 7.2 Lyapunov’sMethod ................................ ...... 36 8 Laplace Transform Method 39 8.1 Transformations ................................. ....... 39 8.1.1 LaplaceTransform .............................. ..... 39 8.1.2 PropertiesoftheLaplaceTransform . .......... 39 8.1.3 Laplace Transform Method for solving Initial Value Problems........... 40 8.1.4 InverseLaplaceTransform. ........ 40 8.1.5 Convolution: .................................. 41 8.1.6 StepFunction.................................. 41 8.1.7 Impulseresponse ............................... ..... 41 9 Numerical Methods 43 9.1 NumericalMethods................................ ....... 43 9.1.1 Euler’sMethod................................. 43 9.1.2 ImprovedEuler’sMethod . ...... 43 9.1.3 Including the Second Order Term in Taylor’s Expansion .............. 44 9.1.4 OtherMethods .................................. 44 A Brief Review of Complex Numbers 45 A.1 Introduction.................................... ....... 45 A.2 Euler’s Formula and Exponential Representation . ................ 46 B Similarity Transformation and the Jordan Canonical Form 49 B.1 SimilarityTransformation . ........... 49 B.2 Diagonalization................................. ........ 49 B.3 JordanForm ...................................... 49 Chapter 1 Introduction to Differential Equations 1.1 Introduction: Many phenomena in engineering, physics and broad areas of applied mathematics involve entities which change as a function of one or more variables. The movement of a car along a road, the propagation of sound and waves, the path an airplane takes, the amount of charge in a capacitor in an electrical circuit, the way a cake taken out from a hot oven and left in room temperature cools down and many such processes can all be modeled and described as differential equations. In class we discussed a number of examples such as the swinging pendulum, a falling object with drag and an electrical circuit involving resistors and capacitors. Let us make a definition: Definition 1.1.1 A differential equation is an equation that relates an unknown function and one or more of its derivatives of with respect to one or more independent variables. For instance, the equation dy = 5x dx − relates the first derivative of y with respect to x, with x. Here x is the independent variable and y is the unknown function. In this course, we will learn how to solve such differential equations. We will make a precise definition of what solving a differential equation means shortly. dny Before proceeding further, we make a remark on notation. Recall that dxn is the n-th derivative of y (n) dny with respect to x. One can also use the notation y to denote dxn . It is further convenient to write y′ = y(1) and y′′ = y(2). In physics, the notation involving dots is also common, such thaty ˙ denotes the first-order derivative. Different classifications of differential equations are possible and such classifications make the analysis of the equations more systematic. 1.2 Classification of Differential Equations There are various classifications for differential equations. Such classifications provide remarkably simple ways of finding the solutions (if they exist) for a differential equation. Differential equations can be of the following classes: 7 1.2.1 Ordinary Differential Equations If the unknown function depends only on a single independent variable, such a differential equation is ordinary. The following is an ordinary differential equation: d2 d 1 L Q(t)+ R Q(t)+ Q(t)= E(t), dt2 dt2 C which is an equation which arises in electrical circuits. Here, the independent variable is t. 1.2.2 Partial Differential Equations If the unknown function depends on more than one independent variables, such a differential equation is said to be partial. Heat equation is an example for partial differential equations: ∂2 ∂ α2 f(x,t)= f(x,t) ∂x2 ∂t Here x,t are independent variables. 1.2.3 Homogeneous Differential Equations If a differential equation involves terms all of which contain the unknown function itself, or the deriva- tives of the unknown function, such an equation is homogeneous. Otherwise, it is non-homogeneous. 1.2.4 N-th order Differential Equations The order of an ordinary differential equation is the order of the highest derivative that appears in the equation. For example 2y(2) + 3y(1) = 5, is a second-order equation. 1.2.5 Linear Differential Equations A very important class of differential equations are linear differential equations. A differential equation F (y, y(1), y(2),...,y(n))= g(x) is said to be linear if F is a linear function of the variables y, y(1), y(2),...,y(n). Hence, by the properties of linearity, any linear differential equation can be written as follows: n (i) an−i(x)y = g(x) Xi=0 for some a0(x),a1(x),...,an(x) which don’t depend on y. For example, ay + by(1) + cy = 0 is a linear differential equation, whereas yy(1) + 5y = 0, is not linear. If a differential equation is not linear, it is said to be non-linear. 1.3 Solutions of Differential equations Definition 1.3.1 To say that y = g(x) is an explicit solution of a differential equation dy dny F (x,y, , ) = 0 dx dxn on an interval I R, means that ⊂ dg(x) dng(x) F (x, g(x), ,..., ) = 0, dx dxn for every choice of x in the interval I. Definition 1.3.2 We say that the relation G(x, y) = 0 is an implicit solution of a differential equation dy dny F (x,y, , ) = 0 dx dxn if for all y = g(x) such that G(x, g(x)) = 0, g(x) is an explicit solution to the differential equation on I. Definition 1.3.3 An n-th parameter family of functions defined on some interval I by the relation h(x,y,c1,...,cn) = 0, is called a general solution of the differential equation if each member of the family is an explicit solution. Each element of the general solution is a particular solution. Definition 1.3.4 A particular solution is imposed by supplementary conditions that accompany the differential equations. If all supplementary conditions relate to a single point, then the condition is called an initial condition. If the conditions are to be satisfied by two or more points, they are called boundary conditions. Recall that in class we used the falling object example to see that without a characterization of the initial condition (initial velocity of the falling object), there exist infinitely many solutions. Hence, the initial condition leads to a particular solution, whereas the absence of an initial condition leads to a general solution. Definition 1.3.5 A differential equation together with an initial condition (boundary conditions) is called an initial value problem (boundary value problem). 1.4 Direction Fields The solutions
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